Finite generation properties of the pro-$p$ Iwahori-Hecke $\operatorname{Ext}$-algebra
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913339289894912 |
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| author | Bodon, Emanuele |
| author_facet | Bodon, Emanuele |
| contents | The pro-$p$ Iwahori-Hecke $\operatorname{Ext}$-algebra $E^\ast$ is a graded algebra that has been introduced and studied by Ollivier-Schneider, with the long-term goal of investigating the category of smooth mod-$p$ representations of $p$-adic reductive groups and its derived category. Its $0$th graded piece is the pro-$p$ Iwahori-Hecke algebra studied by Vignéras and others.
In the present article, we first show that the $\operatorname{Ext}$-algebra $E^\ast$ associated with the group $\operatorname{SL}_2(\mathfrak{F})$, $\operatorname{PGL}_2(\mathfrak{F})$ or $\operatorname{GL}_2(\mathfrak{F})$, where $\mathfrak{F}$ is an unramified extension of $\mathbb{Q}_p$ with $p \neq 2,3$, is finitely generated as a (non-commutative) algebra.
We then specialize to the case of the group $\operatorname{SL}_2(\mathbb{Q}_p)$, with $p \neq 2,3$, and we show that in this case the natural multiplication map from the tensor algebra $T^\ast_{E^0} E^1$ to $E^\ast$ is surjective and that its kernel is finitely generated as a two-sided ideal. Using this fact as main input, we then show that $E^\ast$ is finitely presented as an algebra. We actually compute an explicit presentation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00916 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite generation properties of the pro-$p$ Iwahori-Hecke $\operatorname{Ext}$-algebra Bodon, Emanuele Representation Theory Number Theory 20C08, 22E50, 16S15, 16E30, 20J06, 20G25, 11F85 The pro-$p$ Iwahori-Hecke $\operatorname{Ext}$-algebra $E^\ast$ is a graded algebra that has been introduced and studied by Ollivier-Schneider, with the long-term goal of investigating the category of smooth mod-$p$ representations of $p$-adic reductive groups and its derived category. Its $0$th graded piece is the pro-$p$ Iwahori-Hecke algebra studied by Vignéras and others. In the present article, we first show that the $\operatorname{Ext}$-algebra $E^\ast$ associated with the group $\operatorname{SL}_2(\mathfrak{F})$, $\operatorname{PGL}_2(\mathfrak{F})$ or $\operatorname{GL}_2(\mathfrak{F})$, where $\mathfrak{F}$ is an unramified extension of $\mathbb{Q}_p$ with $p \neq 2,3$, is finitely generated as a (non-commutative) algebra. We then specialize to the case of the group $\operatorname{SL}_2(\mathbb{Q}_p)$, with $p \neq 2,3$, and we show that in this case the natural multiplication map from the tensor algebra $T^\ast_{E^0} E^1$ to $E^\ast$ is surjective and that its kernel is finitely generated as a two-sided ideal. Using this fact as main input, we then show that $E^\ast$ is finitely presented as an algebra. We actually compute an explicit presentation. |
| title | Finite generation properties of the pro-$p$ Iwahori-Hecke $\operatorname{Ext}$-algebra |
| topic | Representation Theory Number Theory 20C08, 22E50, 16S15, 16E30, 20J06, 20G25, 11F85 |
| url | https://arxiv.org/abs/2405.00916 |